Using ACF and PACF to Screen Candidate Orders

Why ACF and PACF Patterns Help Screen Orders

When a time series has been made suitable for stationary modeling, its dependence across time can help narrow the plausible model orders. The autocorrelation function (ACF) summarizes correlation between observations separated by each lag. The partial autocorrelation function (PACF) measures the correlation at a lag after accounting for the intervening lags. Their patterns can suggest AR or MA structure, but sample plots are screening tools: finite samples fluctuate, and real series need not display ideal theoretical cutoffs.

The screening should be done on a series whose trend, level changes, and seasonality have been considered. A persistent trend can make many ACF values large even when the short-memory order is small. If a preliminary trend or seasonal component has been removed, the ACF and PACF of the resulting residuals can help expose remaining dependence. They do not, by themselves, establish that the preliminary treatment or a proposed model is adequate.

A common sample ACF definition, using observations \(x_1,\ldots,x_n\) and their sample mean \(\bar{x}\), is

In particular, \(r_0=1\). Software can use slightly different finite-sample conventions, so exact plotted values may differ a little. The PACF at lag \(k\) is commonly estimated by fitting the linear prediction of \(X_t\) from its preceding \(k\) values and reading the coefficient on \(X_{t-k}\). At lag 1, the PACF equals the ACF.

Theoretical Patterns and Their Limits

For a stationary autoregressive model of order \(p\), written AR(\(p\)), the theoretical PACF is zero beyond lag \(p\), while the ACF generally tails off. For an invertible moving-average model of order \(q\), written MA(\(q\)), the ACF is zero beyond lag \(q\), while the PACF generally tails off. A stationary, invertible ARMA model usually has both functions tail off rather than showing a clean cutoff. “Tails off” may mean gradual decay, possibly with alternating signs or damped oscillation; it does not require a smooth monotone decline.

These are population patterns under the stated model conditions, not rules that every sample plot must obey. A finite-sample ACF may have a nonzero value past a theoretical cutoff, and a true nonzero correlation may look small in a short series. If the process is nonstationary, these stationary-model patterns are not a sound basis for selecting orders until the nonstationarity has been addressed.

One reason for the MA cutoff follows directly from the model. Suppose \(X_t=\varepsilon_t+\theta\varepsilon_{t-1}\), where the innovations \(\varepsilon_t\) are uncorrelated, have mean zero, and have variance \(\sigma^2\). Then

For \(k\geq 2\), \(X_t\) and \(X_{t-k}\) contain no common innovation, so their covariance is zero. Dividing each covariance by \(\gamma_0\) gives

That is the theoretical ACF cutoff for an MA(1). In observed data, the estimated \(r_k\) values beyond lag 1 will generally not be exactly zero. By contrast, for a stationary AR(1), the ACF follows \(\rho_k=\phi^k\), where \(|\phi|<1\), so it decays rather than cutting off; its PACF cuts off after lag 1.

Reference Bands: A Useful Approximation, Not a Verdict

A standard visual reference for an ACF under a white-noise null is the approximate pointwise band

Under white noise and suitable large-sample conditions, an individual sample autocorrelation at a fixed nonzero lag has approximate standard error \(1/\sqrt{n}\). The factor 1.96 gives an approximate 95% pointwise reference interval. The same rough band is often used when scanning PACFs, although more careful uncertainty calculations can depend on the setting and estimation method.

These bands are not simultaneous confidence bands for all lags. When many lags are inspected, some estimates may fall outside by chance. Also, if the series has meaningful autocorrelation, the white-noise approximation need not describe the uncertainty at later lags. Use the band to judge whether an isolated value is striking relative to sampling noise, not as a mechanical keep-or-reject test.

Seasonal Lags and Preliminary Residuals

For data observed at a regular seasonal interval \(s\), inspect not just the first few lags but also \(s,2s,3s,\ldots\). For monthly data with annual seasonality, these are lags 12, 24, 36, and so on. For quarterly data with annual seasonality, they are lags 4, 8, 12, and so on. Repeated dependence at those lags can indicate that seasonal structure remains. A seasonal AR pattern may show a PACF feature at a seasonal lag and a declining ACF at seasonal multiples; a seasonal MA pattern may instead show an ACF feature at a seasonal lag. Mixed or short samples can blur these ideal patterns.

A residual ACF is useful only with context: identify which trend, seasonal, or other preliminary structure was removed before calculating it. For example, residual autocorrelation at lags 12 and 24 after removing a monthly series’ trend may signal that seasonal dependence remains. Conversely, residual values near zero at those lags are not proof of adequate seasonal modeling. Later model comparisons and residual diagnostics are needed to assess a fitted candidate.

Common Trap: Calling a Sample Cutoff Exact

Suppose an analyst sees a PACF bar outside a reference band at lag 1 and no later bars outside it. The incorrect conclusion is, “The series is definitely AR(1), so every other order is ruled out.” The correction is, “The pattern supports AR(1) as a candidate, subject to model comparison and further checking.” A sample PACF can hide smaller true effects, and an ARMA process can resemble a pure AR process over a limited range of lags.

A related calculation error is to change the denominator of the common white-noise band at every lag. For \(n=64\), an incorrect lag-6 band would use \(1.96/\sqrt{64-6}=1.96/\sqrt{58}\approx0.257\). The usual rough white-noise pointwise band is instead \(1.96/\sqrt{64}=0.245\) at each fixed lag. More importantly, neither version makes a visual cutoff conclusive; the band is approximate and pointwise.

Other practical cautions are to avoid reading a trending raw series as a short-memory AR model, to inspect seasonal multiples as well as small lags, and to avoid selecting a high order merely because one late-lag bar crosses a band. ACF and PACF plots are most useful when they narrow a small, defensible set of candidates.